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Theorem xmstopn 23710
Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
isms.j 𝐽 = (TopOpen‘𝐾)
isms.x 𝑋 = (Base‘𝐾)
isms.d 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
xmstopn (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Proof of Theorem xmstopn
StepHypRef Expression
1 isms.j . . 3 𝐽 = (TopOpen‘𝐾)
2 isms.x . . 3 𝑋 = (Base‘𝐾)
3 isms.d . . 3 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
41, 2, 3isxms 23706 . 2 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
54simprbi 497 1 (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105   × cxp 5618  cres 5622  cfv 6479  Basecbs 17009  distcds 17068  TopOpenctopn 17229  MetOpencmopn 20693  TopSpctps 22187  ∞MetSpcxms 23576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-xp 5626  df-res 5632  df-iota 6431  df-fv 6487  df-xms 23579
This theorem is referenced by:  imasf1oxms  23751  ressxms  23787  prdsxmslem2  23791  tmsxpsmopn  23799  xmsusp  23831  cmetcusp1  24623  minveclem4a  24700  minveclem4  24702  qqhcn  32239  rrhcn  32245  rrexthaus  32255  dya2icoseg2  32545  sitmcl  32618
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